41
Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts1
Ulrich Haskamp2
Abstract
There is empirical evidence for a time-varying relationship between
exchange rates and fundamentals. Such a relationship with time-varying
coefficients can be estimated by a Kalman filter model. A Kalman filter
estimates the coefficients recursively depending on the prediction error of the
examined model. Using a Taylor rule based exchange rate model, which in the
literature was found to have promising forecasting abilities, it is possible to
further improve the performance if the utilization of information from the
prediction error is restricted. This is necessary as the bad performance of
classic exchange rate models is not solely due to their neglect of the time
varying relationship, but also due to missing explanatory information. So, if
the Kalman filter used the entire information from the prediction error, it
would overestimate the need for coefficient adjustment. With this calibration
of the Kalman filter model the short-term out-of-sample forecasting accuracy
can be enhanced for 10 out of 12 exchange rates compared to an OLS
estimation. Without our adjustment of the calibration, the Kalman filter
performs worse.
Key words: Exchange rates, forecasting, Kalman filter, state space model
JEL Classification: C35, F31, F37
1 This project has received funding from the European Union’s Seventh Framework
Programme for research, technological development and demonstration under grant agreement
no. 612955. 2 Ruhr Graduate School in Economics, Kiel University and University of Duisburg-Essen,
Universitätsstr. 12, 45117 Essen, Germany, [email protected].
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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1 - Introduction
Exchange rates are important for many economic agents. Currencies
need to be exchanged when goods are traded across borders, investors
finance projects abroad or emigrants send parts of their wage home. In an
increasingly globalized world this is of particular importance. However, all
these economic agents face exchange rate risks. The costly consequences of
these risks could be mitigated by reliable exchange rate forecasts.
Performing reliable exchange rate forecasts is, however, one of the puzzles
of the economic profession. Meese and Rogoff (1983) found in their seminal
paper that classic exchange rate models were not able to forecast better than
a naive random walk. Cheung et al. (2005) confirmed this finding for newer
models and a longer sample.
A possible reason for the non-satisfactory performance of economic
exchange rate models is that the relationship between exchange rates and their
fundamentals varies over time. For example, if the output of a country
decreases by just a small amount, this generally does not make a significant
impact on the country’s exchange rate. In times of recession, however, the
impact of exchange rate variation is much stronger. In such a situation the
output is likely to have a stronger impact on the exchange rate than usual.
Another reason for a time-varying relationship is provided by the scapegoat
model of Bacchetta and van Wincoop (2004). In such a model the exchange
rate can change due to unobservable factors such as non-speculative trading,
but investors attribute this change to other observable explanatory variables
which increases their impact on the exchange rate.
Furthermore, empirical evidence for such a relationship can be found
in the literature. Fratzscher et al. (2015) tested the aforementioned scapegoat
theory empirically using survey data of about 50 foreign exchange investors
who were asked about their individual weighting of the importance of various
fundamentals for 12 currencies. Additionally, they obtained order flow data
from the UBS, a major player in the foreign exchange market, to model the
unobservable factors that drive exchange rates. They found that the in-sample
performance of an exchange rate model for the period of 2001-2009 including
this time-varying weighting and the unobservable factors improves
substantially for all examined currency pairs. Further in-sample evidence is
provided by Beckmann et al. (2011), who confirmed that the relationship
between the German mark/euro-US dollar exchange rate and its fundamentals
incorporates breaks and depends on different fundamentals over time. Out-of-
sample evidence is provided by Sarno and Valente (2009). They show, using a
rolling forecasting exercise, that the best model to forecast exchange rates
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
43
consists of frequently changing fundamentals. In this paper we estimate a
time-varying relationship and use it then for out-of-sample forecasting, as
done by Rossi (2006).
For the underlying exchange rate model, a Taylor rule based model
which was found to have promising out-of-sample forecasting abilities for
the short-term is used (Rossi, 2013). Additionally, contrary to Meese and
Rogoff (1983), for example, true out-of-sample forecasts are performed.
True out-of-sample forecasts means that no rational expectations about
future realizations of the explanatory variables are needed, and only lagged
variables are used. Such Taylor rule models were first used for forecasting
by Molodtsova and Papell (2009). They evaluated such models as more
useful than classic exchange rate models as the purchasing power parity or
the monetary approach. A time-varying relationship can be inserted into the
models using a Kalman filter. The Kalman filter approach is chosen for two
reasons: Firstly, this framework allows for continuously changing
coefficients which are not restricted to certain historical or recurring values.
Secondly, a stochastic component is embedded. Those features justify our
choice ahead of other techniques such as Markov switching or smooth
transition models. As in Schinasi and Swamy (1987), Wolff (1987), Ng and
Heidari (1992) or Rossi (2006), it is assumed that the coefficients of the
exchange rate model behave like random walks and evolve over time. The
Kalman filter allows a recursive estimation of the unobservable time-varying
variation in the coefficients. Using an updated version of the data set of
Molodtsova and Papell (2009), it is then tested whether this modification
improves the forecasting performance compared to the baseline model.
The starting point for our calibration of the Kalman filter is Rossi
(2006). The coefficients are assumed to behave as a random walk. The
evolution of the time-varying coefficients is slowed down by introducing a
parameter smaller than one into the state equations’ error term of the Kalman
filter model. Our newly introduced feature is that also the considered
prediction error, which usually is very high for exchange rate models (Evans
and Lyons (2002)), is reduced by incorporating another parameter lower than
one into the signal equation’s error term. So, if the performance of exchange
rate models is weak not only because of non-time-varying coefficients, but
also due to missing information, then the Kalman filter would overestimate
the prediction error and thereby the need for coefficient adjustment. With this
calibration the short-term forecasting accuracy can be enhanced for 10 out of
12 exchange rates. This favorable result is also robust against other window
sizes of the rolling regressions. This is interesting as for lower window sizes a
time-varying relationship is already more strongly implied for the baseline
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
44
model. Comparisons against the random walk without drift show that the
result of Meese and Rogoff (1983) prevails.
This paper’s contribution is to show that applying a time-varying
coefficient approach to Taylor rule exchange rate models can further improve
the short-term out-of-sample forecasting abilities compared to simple OLS
forecasts. Using a very long sample, examining 12 different exchange rates
and varying window sizes, the result is found to be robust. However, this is
only possible if one adjusts the calibration of the Kalman filter model for the
estimation of the underlying exchange rate model. This opposes the results of
older papers such as Schinasi and Swamy (1987), Wolff (1987) or Ng and
Heidari (1992) who found, using short samples of about 10 years, that a
simple Kalman filter model is better in forecasting exchange rates than OLS.
Other authors such as Rossi (2006), using longer and more recent samples, did
not obtain such results. In our robustness chapter we also find that the nearer
we get to the standard calibration of the Kalman filter, the worse the results
become. Therefore, our new calibration adds value to the literature by
showing that a Kalman filter can be capable of improving exchange rate
forecasts - also for more recent data.
The outline of this work is as follows: In the first section the
underlying Taylor rule exchange rate model, as used by Molodtsova and
Papell (2009), is stated. Secondly, it is explained how the time-varying
relationship is incorporated into the exchange rate model with a Kalman filter
model. After this, the criteria to evaluate the forecasting performance, the data
set and the forecasting methodology are described. Then, the results of a
forecast performance comparison between a Taylor rule exchange rate model
including a time-varying relationship and a benchmark model, either the
baseline Taylor rule model of Molodtsova and Papell (2009) or the random
walk, are presented. Lastly, the robustness of the previous results with regard
to different window sizes of the rolling regressions and the Kalman filter
calibration is examined.
2 - Taylor Rule Exchange Rate Model
The underlying exchange rate model for the forecast comparison
between a model with and without a time-varying relationship is Taylor
(1993) rule based. Such Taylor rule models have been applied to exchange
rate economics by Engel and West (2005), Engel and West (2006) or Mark
(2009). Engel et al. (2007), Wang and Wu (2012) and Lansing and Ma (2015)
found that they are useful for exchange rate forecasting. Additionally,
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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Molodtsova et al. (2008), Molodtsova and Papell (2009), and Beckmann and
Schüssler (2016) noted that the influence of the included variables varies over
time. Also, the Taylor rule model itself has been estimated with time-varying
coefficient model. See Yüksel (2013) for a survey.
Taylor’s rule relies on the idea that a central bank sets its short-term
key interest rate as a function of inflation, targeted real key interest rate and
output gap:
(1)
An asterisk denotes the targeted level. This equation shows that the central
bank raises its key interest rate target, , in case of inflation, , rising above
the targeted level. It also sets the rate higher when the output gap, ,
increases. Because it is assumed that the target for the key interest rate is
reached within one period, the actual interest rate is not included (Mishkin,
2004).
From this basic equation, we can get the nominal interest rate reaction
function assuming that the central bank targets its currency’s purchasing
power parity (PPP) to hold and that adjustments to the interest rate are only
done successively. For details refer to Molodtsova and Papell (2009). So, the
derived equation looks like the following:
(2)
The represents the real exchange rate. For the United States is equal to
zero because it is the reference currency for all exchange rates examined
later. To obtain an estimable equation, the US interest rate reaction function
is subtracted by its foreign country’s counterpart:
(3)
A tilde denotes a foreign variable. The indices and display coefficients of
the US and the foreign country, respectively. is a constant, and for both countries, and for the foreign
one. The error term of this two-country equation is denoted as .
The effect of a successively adjusted interest rate can be explained in the
following way: If inflation rises, the central bank will increase the interest rate
by in the first period. In the second period, it will be raised by
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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and so forth. Note that is assumed to be between zero and
one. If one assumes uncovered interest rate parity (UIP), the interest rate
differential is proportional to the expected exchange rate depreciation:
(4)
Inserting this equation into Equation (3) and assuming rational expectations, a
Taylor rule based exchange rate forecasting equation is obtained:
(5)
However, the literature shows that the UIP does not hold in the short run.
Chinn (2006) examines new empirical findings about the UIP and found that
in the short-term the interest rate differential is still a biased predictor of ex
post changes of exchange rates. Cheung et al. (2005) found that the UIP can
forecast exchange rates only in the long-run. Recent carry trade literature,
such as Burnside et al. (2007), even suggests that of Equation (4) can be
negative. Still, this equation can be used for forecasting if the signs and values
of the coefficients are not restricted.
Furthermore, Molodtsova and Papell (2009) made the prediction that
an increase of the interest rate results in exchange rate appreciation.
Gourinchas and Tornell (2004) presented a theoretical model in which the
investors misperceive the persistence of an interest rate change. This allows
for a mechanism in which an increase of the interest rate leads to an exchange
rate appreciation. Additionally, they presented survey evidence which shows
that investors underestimate the persistence of interest rate shocks to support
their theory.
If interest rate smoothing is assumed, higher inflation not only raises
the current interest rate, but also leads to the expectation that the interest rate
will continue to rise in the future. Therefore, an increase of the inflation rate
does result in immediate and expected future exchange rate appreciation. A
further effect of interest rate smoothing, in the context of the Gourinchas and
Tornell (2004) theory, is that investors are even more likely to underestimate
the true nature of persistence of interest rate shocks because the initial change
of the interest rate is smaller than the total change over time. This strengthens
the effect of inflation on the exchange rate (Molodtsova and Papell, 2009).
Three additional predictions in the context of the Taylor rule are made
by Molodtsova and Papell (2009). Firstly, an increase in the US output gap
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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leads to a rise of the interest rate which then results in a depreciation of the
exchange rate. Secondly, if the real exchange rate depreciates, the foreign
central bank will raise its interest rate to revert the real exchange rate back to
its PPP equilibrium. Lastly, assuming interest rate smoothing, a higher lagged
interest rate leads to further immediate and expected future depreciation of the
exchange rate.
This Taylor rule forecasting equation can be used in several different
specifications of the variables on the right-hand side. Firstly, in his original
paper Taylor (1993) stated for Equation (3) that the central bank sets the
nominal interest rate only according to the inflation gap, the output gap and
the equilibrium real interest rate omitting the lagged interest rate differential
and the real exchange rate. If the foreign central bank behaves similarly, this
model is called symmetric. However, following Clarida et al. (1998), it can be
assumed that the foreign central bank aims at the difference between the
actual and targeted exchange rate defined by PPP. In such an asymmetric
model the real exchange rate is included in the equation, . Here,
however, only symmetric models, for which Molodtsova and Papell (2009)
found the best predictability, are examined for conciseness. Alba et al. (2015)
examine asymmetric models for inflation-targeting emerging markets for
which the real exchange rate is likely to be an important determinant.
The next specification adds the lagged interest rate differential to the
right- hand side to account for a successively adjusted interest rate, as
suggested by Clarida et al. (1998). In this case the model is denoted as with
smoothing, , or otherwise as with no smoothing.
Lastly, if both central banks respond equally to changes of the right-
hand side variables so that , and , the model is
called homogeneous. If they are not identical, each appears solely on the right-
hand side and the model is called heterogeneous. Therefore, the models which
are considered are with or without smoothing and include homogeneous or
heterogeneous coefficients.
3 - Modeling the Time-Varying Relationship
Section 1 outlined empirical evidence for a time-varying relationship
between exchange rates and fundamentals. In this section the estimation of
such a relationship with a Kalman (1960) filter model is described. Other
authors, for example, Canova (1993), Carriero et al. (2009) or Beckmann and
Schüssler (2016) used other models which allow for a time-varying
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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relationship. See Rossi (2013) for a further review.
First of all, a short introduction to a Kalman filter model is given. A
Kalman filter model is a state space model which consists of signal and state
equations. The signal equation describes the relationship between the
observed and unobserved variables. The state equations represent the
evolution of the unobserved variables, here the time-varying coefficients, .
The Kalman filter recursively estimates the unobserved variables conditional
on the available information. With some chosen starting values the first signal
equation is estimated in . In the prediction error is calculated by
comparing the actual dependent variable with the estimated one. This
prediction error is then used to update the beliefs about . Using the
updated coefficient the next signal equation is estimated and so forth.
Kalman filter models with a time-varying relationship have already
been used in exchange rate forecasting. Schinasi and Swamy (1987) allowed
the coefficients of their monetary exchange rate models to vary over time in
the following way:
(6)
(7)
(8)
(9)
The first equation of this system is the signal equation with containing the
explanatory variables. Equation (7) is the state equation which describes the
development of the coefficients. As the error term of the state equation is
assumed to be of autoregressive order, the coefficients follow an
autoregressive process. Comparing their forecasts with the random walk, they
found favorable results if a lagged dependent variable is included.
Wolff (1987) used the Kalman filter to estimate such a time-varying
parameter model for similar monetary models. Contrary to Schinasi and
Swamy (1987), he assumed that the coefficients behave as a random walk so
that . He found that such a model is often able to forecast more
accurately than the random walk. However, a newer study by Rossi (2006)
was not able to obtain similar findings for autoregressive models, with or
without further economic variables, and using a longer sample. Rossi (2013),
giving an overview of further studies, comes to the conclusion that the
empirical evidence is mixed. Furthermore, Bacchetta et al (2010) conclude
that time-varying parameters cannot explain the puzzle of Meese and Rogoff
(1983) which, though, is debated by Chinn (2009) and Giannone (2009).
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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t
Here the time-varying relationship between exchange rates and
fundamentals is modeled comparable to Wolff (1987) and Rossi (2006).
However, it is newly applied to models with Taylor rule fundamentals for
short-term forecasting and a longer sample.
(10)
(11)
(12)
This model can be formulated more concisely:
(13)
(14)
where
and with vector including the fundamentals.
Thus, the Kalman filter is calibrated in the following way: Starting
values for the coefficients are set to zero for every regression. For all state
equations the error terms are assumed to be equal, and are normalized by a
covariance matrix which is specified by
. Furthermore,
is the ratio between the variance of the signal and state equations’ error
terms. A smaller leads to a slower coefficient evolution. Until now the
calibration of the Kalman filter is the same as in Rossi (2006) or Stock and
Watson (1996).
The difference to previous approaches is that a parameter is
inserted into the signal equation’s error term. The effect is that the prediction
error, , which usually is very high in exchange rate models (Evans
and Lyons (2002)), has less of an effect on the updating of the coefficients. If
the bad performance of exchange rate models is not solely due to non-time-
varying coefficients, but also due to missing explanatory information, then the
Kalman filter would adjust the coefficients too strongly. Henceforth, is set
to and is set to . The value of is chosen to be very small to
ensure that only a very limited amount of information of the prediction error
is used. The value of is set comparable to Rossi (2006) and slows the
evolution of the coefficients down. In Section 5.1 it will be examined whether
the upcoming results are robust to other calibrations of these parameters. We
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Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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refrain from adding further variables as it is already difficult for a Kalman
filter to estimate a model with 6 variables and account the information of the
prediction error towards a specific variable.
For the upcoming forecast exercise rolling regressions are used. See
Section 4.2 for details. Rolling regressions already implicitly allow
coefficients to change at every step forward. Molodtsova and Papell (2009)
showed in their paper how the coefficients evolve across their sample. Here it
is shown that it is still worthwhile to use models which allow for time-varying
parameters in addition to rolling regressions. In Figure 1 the path of the
coefficients across the sample for rolling regressions estimated with OLS and
a Kalman filter is shown. For the Kalman filter model the final state values
are reported. All equations are estimated for a symmetric Taylor rule model
with smoothing and heterogeneous coefficients for the Pound-US dollar
exchange rate. In general, it can be seen that for both estimation methods the
coefficients follow a similar path, but the coefficients of the Kalman filter
model incorporate more abrupt adjustments. For further explanations about
the path of the coefficients, refer to Molodtsova and Papell (2009).
51
Figure 1: Estimated Coefficients of a Kalman Filter Model and OLS
Recent studies also employed other methods to model the time-
varying relationship. See, for example, Fratzscher et al. (2015) who obtained
actual data about the weighting investors put on fundamentals. Finding
favorable results, their sample is, however, too short to conduct an out-of-
sample forecasting exercise.
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Using Less Information to Obtain Better Forecasts –
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4 - Comparison of Forecast Performance
In this section the forecasting performance of the Taylor rule models
will be examined with a special focus on whether it is possible to improve
forecasting by taking the time-varying relationship between exchange rates
and fundamentals into account.
1.1 Evaluating Forecasting Performance
The forecasting performance of the models is evaluated by
calculating the mean squared prediction error (MSPE). For a sample with
observations the MSPE of forecasts is calculated as follows:
(15)
Where is the one-period ahead forecast error of the respective model
. The forecast for the random walk is that the exchange rate does not
change, . With the Diebold-Mariano test (1995), it is
possible to test whether the MSPE of a model is different from a competing
model. The null hypothesis is that both models’ MSPE are equal which is
tested against the alternative that model has a higher MSPE:
(16)
(17)
For the construction of the Diebold-Mariano (DM) test statistic, the average
difference between the forecasts’ errors of the competing models is defined
as:
(18)
Then the DM test statistic can be computed as follows:
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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(19)
The denominator represents the standard error of the difference between the
competing model’s forecast errors. The test statistic of the Diebold-Mariano
test can easily be gained by regressing the difference of the forecast errors on
a constant. To estimate the standard error consistently, a Newey-West
covariance estimator which is consistent in the presence of both
heteroskedasticity and autocorrelation is used (Elliot et al., 2006). For the
covariance estimator the lag length is chosen by the automatic bandwidth
selection method for kernel estimators of Newey-West (1994). Because the
DM statistic has an asymptotic normal distribution, the resulting t-statistic of
the regression can be used to test the above mentioned hypothesis (Diebold
and Mariano, 1995). Note that the DM test in this setup is one-sided.
Therefore, a significant and positive DM test statistic implies that model
outperforms because is significantly higher than
. The critical
values for a one-sided test and are 1.282, 1.645 and 1.960 for the
significance levels 0.10, 0.05 and 0.01, respectively.
Newer test statistics for a test which compares nested models also
exist and were used, for example, by Molodtsova and Papell (2009),
Gourinchas and Rey (2007) or Engel et al. (2007). In nested models, one
model is contained within the other. This is, for example, given for a
comparison between the random walk and any other model. Note that this is
not the case if Taylor rule models with and without time-varying
relationships are compared as the models are different due to the time-
varying coefficients. The tests of Clark and West (2006, 2007) and Clark and
McCracken (2001, 2005) compensate for the fact that the DM test statistic is
undersized under the null hypothesis when comparing nested models. In the
context of exchange rates, both tests take into account that, given the null
hypothesis, the exchange rate follows a random walk. However, as Rogoff
and Stavrakeva (2008) showed, these tests are no real minimum MSPE tests.
Rather, they show if the true model is a random walk instead of whether the
random walk has a lower MSPE than the structural model. So, a DM test
compares forecast ability while the newer tests look at predictability
(Molodtsova and Papell, 2009). Here it is stuck to the DM test for a
Ulrich Haskamp – Improving Exchange Rate Forecasting with a Kalman Filter:
Using Less Information to Obtain Better Forecasts –
Frontiers in Finance and Economics – Vol 13 N°2, 41-73
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comparison between nested models.
4.2 - Data Set and Forecasting Methodology
The data used in this work was obtained from David Papell’s
homepage and has been updated. Besides the update it is the same as in
Molodtsova and Papell (2009).3 It ranges, in general, from March 1973 to
December 1998 for the euro area countries and to May 2015 for the others.4
Due to missing data, the sample starts in January 1975 for Canada,
September 1975 for Switzerland and January 1983 for Portugal. The main
source for the data is the International Monetary Fund’s International
Financial Statistics database.5 Because GDP data is available only quarterly,
a seasonally adjusted monthly industrial production index is used for the
calculation of the output gap. The economies’ price levels are measured by
the consumer price index and inflation is then calculated as the 12-month
difference of the price index. For the interest rates set by the central banks
the money market rates are taken. The exchange rates are obtained from the
Federal Reserve Bank of Saint Louis’ database.
For the construction of the output gap, a Hodrick and Prescott (1997)
filter with the standard smoothing parameter of 14,400 for monthly
frequencies was used, with a sample beginning in January 1971, to obtain
the potential output. The output gap is then calculated by the percentage of
deviation of the potential and actual output. Because rolling regressions are
used for forecasting, the output gap is recalculated for each step with data
excluding the forecasted month. So, no ex post data is used for the
calculation. Thereby, it is simulated that the central bank decision to set the
interest rate is based only on the information available at that time. However,
this only mimics real-time data because revised data is still used. Another
shortcoming is that it is not known how exactly the central banks calculate
the output gap for their decision making. For example, Orphanides (2001),
estimating Taylor rules for the USA, used output gap data generated by the
US central bank itself, but such data is not available for the full sample or
3 Compare http://www.uh.edu/˜dpapell/papers2.htm.
4 The euro area currencies are French franc, German mark, Italian lira,
Dutch guilder and Portuguese escudo. The others are Canadian dollar,
Australian dollar, Danish kroner, Japanese yen, Swedish kronor, Swiss
franc and British pound. 5 The only exception is that the German consumer price index is taken from
the OECD Main Economic Indicators database.
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other countries. Nevertheless, Orphanides and van Norden (2002) stated that
most of the difference between revised and real-time data of output gap
estimation comes from using ex-post data, not from revisions.
With this data the Forecasting Equations (5) and (10) are estimated.
The regressions are then used for the one-month ahead forecasts of the
exchange rates. As in Molodtsova and Papell (2009), a sample of 120
observations is taken for the rolling regressions. Thus, the first estimation is
done with data from March 1973 until February 1983 and the exchange rate
is out-of-sample fore- casted for March 1983. The out-of-sample forecast is
done with data for the explanatory variables excluding the forecast period
contrary to Meese and Rogoff (1983). Thus, rational expectations about their
development are not needed and the forecasts are truly out-of-sample. For
the next estimation the first data point is dropped and an additional one is
added at the end of the sample. Then the next out-of-sample forecast
follows. So, the regressions roll over the full sample. The full sample,
however, differs for the countries. As previously mentioned, the euro area
countries’ samples end in December 1998 and data is missing for Portugal,
Switzerland and Canada. So, for the euro area exchange rates 190 instead of
388 forecasts are performed, for Portuguese, Swiss and Canadian exchange
rates only 72, 358 or 365, respectively. Note that also the output potential is
calculated again with each step, but the beginning of the sample is not
dropped.
Because the various specifications for Equations (5) and (10) differ
in their forecasting performance, only symmetric specifications for which
Molodtsova and Papell (2009) found the best predictability are evaluated.
First, the MSPE for each exchange rate is calculated. Secondly, the Diebold-
Mariano test statistic for a comparison between the model of Equation (10)
and the benchmark model, which is either the random walk or the
corresponding model without a time-varying weighting, Equation (5), is
determined.
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Using Less Information to Obtain Better Forecasts –
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Figure 2: Forecast Comparison between Equation (5) and (10) for the
Pound- US Dollar Exchange Rate
An example comparison between Equation (5) and (10) for a
symmetric model with interest rate smoothing of the Pound-US dollar
exchange rate is shown in Figure 2. The graph shows the difference between
the forecast errors of both models. A positive value represents that the model
with a time-varying relationship has a smaller forecasting error. The
differences fluctuate around zero and it cannot be said that one model
performs better in a certain period.
4.3 - Results
In this section the results of the forecasting equations are presented.
The finding of Meese and Rogoff (1983), or Cheung et al. (2005) for newer
models and a longer sample, which states that exchange rate models are not
able to forecast better than a random walk without drift which is the toughest
benchmark to beat according to Rossi (2013) still prevails. Therefore, the
focus will be on whether including a time-varying relationship between
exchange rates and fundamentals improves the forecasting performance of
Taylor rule based exchange rate models. In Table 1 the MSPE of symmetric
specifications of the baseline Equation (5) and Equation (10) with a time-
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varying relationship are presented.6 Symmetric models do not include that
the foreign central bank is targeting the real exchange rate. A model denoted
as smoothing incorporates the lagged interest rate. Furthermore, it is
necessary to distinguish between specifications with heterogeneous or
homogeneous coefficients. If a MSPE value is written in bold, then the
model has a MSPE lower than the random walk’s one. This is given for 7 of
the 12 currency pairs. While for Italy most models perform better than the
random walk, this is only given for one of the time-varying ones for Japan,
Sweden and the United Kingdom. For five currencies no specification
forecasts better than the random walk. For all currency pairs, it cannot be
generalized that models with a time-varying relationship perform better.
Additionally, no model is able to significantly beat the random walk as can
be seen in Table 2, which is comparable to the findings of Rogoff and
Stavrakeva (2008) for Taylor rule exchange rate models.
To evaluate whether there is a significant difference between the
Forecasting Equations (5) or (10), Table 3 is presented. The signs of the DM
test statistics vary across currency pairs and Taylor rule model
specifications, but are in general positive. Recall that the critical values for a
one-sided test and are 1.282, 1.645 and 1.960 for the significance
levels 0.10, 0.05 and 0.01, respectively. Therefore, the time-varying Taylor
rule model outperforms the baseline one for 10 of the 12 currencies. For 2
currency pairs no significant statistic was obtained. Furthermore, as the
MSPE is sensitive to outliers, we also use another criterion to evaluate our
forecast. The mean absolute error (MAE) incorporates no squared loss
function and is thus more robust to outliers. We show the results for this
criterion in Table 4. The results are only slightly weaker as we can improve
forecasting only for 9 out of 12 currencies.
In these first results it is shown that estimating a Taylor rule
exchange rate model with a Kalman filter to take time-varying coefficients
into account can improve the forecasting accuracy. Nevertheless, these
improvements are not strong enough to beat the random walk so that the
result of Meese and Rogoff (1983) prevails.
6 The following abbreviations are used henceforth in tables: Can for
Canadian dollar, Aus for Australian dollar, Dan for Danish kroner,
Fra for French franc, Ger for German mark, Ita for Italian lira, Jap
for Japanese yen, Dut for Dutch guilder, Por for Portuguese escudo,
Swe for Swedish kronor, Swi for Swiss franc and UK for British
pound.
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Table 1: MSPE of Symmetric Specifications of Forecasting Equations
(5) and (10)
Specification Can Aus Dan Fra Ger Ita
No
Sm
oo
th.
Heterog. Baseline 1.63 2.76 2.67 2.78 2.92 2.70
Time-varying 1.62 2.76 2.63 2.69 2.84 2.64
Homog. Baseline 1.63 2.76 2.59 2.61 2.84 2.61
Time-varying 1.62 2.73 2.57 2.61 2.79 2.61
Smo
oth
. Heterog. Baseline 1.61 2.78 2.67 2.84 2.94 2.81
Time-varying 1.63 2.74 2.65 2.71 2.86 2.68
Homog. Baseline 1.61 2.78 2.63 2.63 2.87 2.65
Time-varying 1.61 2.73 2.58 2.65 2.81 2.63
Random Walk 1.58 2.71 2.55 2.64 2.75 2.66
Specification Jap Dut Por Swe Swi UK
No
Sm
oo
th.
Heterog. Baseline 2.75 2.78 2.33 2.70 2.92 2.56
Time-varying 2.72 2.76 2.36 2.63 2.91 2.46
Homog. Baseline 2.72 2.77 2.33 2.65 2.77 2.46
Time-varying 2.71 2.74 2.30 2.60 2.78 2.44
Smo
oth
. Heterog. Baseline 2.73 2.85 2.45 2.71 3.01 2.53
Time-varying 2.71 2.77 2.39 2.59 2.95 2.46
Homog. Baseline 2.69 2.80 2.38 2.74 2.80 2.48
Time-varying 2.67 2.75 2.31 2.54 2.82 2.43
Random Walk 2.68 2.75 2.33 2.56 2.74 2.43
The table reports the MSPE for symmetric specifications of the baseline
Equation (5) and (10) with a time-varying relationship. The values were
calculated by forecasts for the months March 1983 to December 1998 for
euro area currencies and to May 2015 for the other ones. Bold values denote
that the model has a lower MSPE than the random walk.
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Table 2: DM Statistics for MSPE Comparisons between Symmetric
Specifications of Forecasting Equations (5) and (10) Against the
Random Walk
Specification Can Aus Dan Fra Ger Ita
No
Sm
oo
th.
Heterog. Baseline -1.51 -0.75 -2.48 -1.73 -1.78 -0.61
Time-varying -1.26 -1.23 -1.75 -0.96 -1.65 0.22
Homog. Baseline -1.96 -1.53 -1.40 0.49 -1.24 0.87
Time-varying -2.06 -0.41 -0.76 0.72 -1.37 0.92
Smo
oth
. Heterog. Baseline -0.90 -0.91 -3.95 -2.00 -1.91 -1.26
Time-varying -1.25 -0.51 -1.75 -0.93 -1.24 -0.24
Homog. Baseline -1.46 -1.65 -2.20 0.05 -1.31 0.10
Time-varying -1.14 -0.53 -0.86 -0.26 -1.56 0.41
Specification Jap Dut Por Swe Swi UK
No
Sm
oo
th.
Heterog. Baseline -1.17 -0.51 0.06 -2.09 -2.87 -1.82
Time-varying -0.73 -0.31 -0.43 -1.65 -3.29 -0.41
Homog. Baseline -1.16 -0.59 -0.03 -2.71 -0.96 -0.75
Time-varying -0.86 0.23 1.28 -1.40 -2.36 -0.25
Smo
oth
. Heterog. Baseline -0.71 -1.32 -0.73 -2.46 -3.52 -1.05
Time-varying -0.45 -0.33 -0.46 -0.40 -4.20 -0.44
Homog. Baseline -0.02 -0.89 -0.41 -2.48 -1.89 -1.03
Time-varying 0.30 0.00 0.46 0.50 -2.93 0.06
The table reports the DM statistics for a test of the equality of forecast
accuracy for symmetric specifications of the baseline Equation (5) and (10)
with a time-varying relationship against the random walk. The values were
calculated by forecasts for the months March 1983 to December 1998 for
euro area currencies and to December 2013 for the other ones. The critical
values for P → are 1.282, 1.645 and 1.960 for the significance levels 0.10,
0.05 and 0.01, respectively.
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Table 3: DM Statistics for MSPE Comparisons between Different
Specifications of Forecasting Equations (5) and (10)
Symmetric Can Aus Dan Fra Ger Ita
No Smooth. Heterog. 0.44 0.01 0.75 1.40 1.38 1.16
Homog. 0.58 1.62 1.21 -0.13 0.91 0.08
Smooth. Heterog. -0.44 1.02 0.40 2.33 1.59 1.82
Homog. 0.29 1.71 1.82 -0.45 0.75 0.26
Symmetric Jap Dut Por Swe Swi UK
No Smooth. Heterog. 1.28 0.52 -0.55 0.92 0.27 2.79
Homog. 0.37 0.86 0.67 2.17 -0.78 1.44
Smooth. Heterog. 1.39 2.15 0.99 1.51 1.31 1.33
Homog. 0.56 1.05 0.64 1.80 -0.69 2.59
The table reports the DM statistics for a test of the equality of forecast
accuracy for symmetric specifications of the baseline Equation (5) and (10)
with a time-varying relationship. The values were calculated by forecasts for
the months March 1983 to December 1998 for euro area currencies and to
May 2015 for the other ones. The critical values for P → are 1.282, 1.645
and 1.960 for the significance levels 0.10, 0.05 and 0.01, respectively.
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Table 4: DM Statistics for MAE Comparisons between Different
Specifications of Forecasting Equations (5) and (10)
Symmetric Can Aus Dan Fra Ger Ita
No Smooth. Heterog. 0.11 1.19 1.29 1.42 1.24 1.41
Homog. -0.01 2.86 0.24 -0.19 0.77 0.19
Smooth. Heterog. -0.23 2.38 0.79 2.56 1.91 2.96
Homog. -0.13 2.55 1.36 -0.40 0.45 0.60
Symmetric Jap Dut Por Swe Swi UK
No Smooth. Heterog. 1.02 1.67 -0.87 1.09 -0.61 2.93
Homog. 0.53 0.65 0.42 1.87 -1.53 0.00
Smooth. Heterog. 1.11 3.07 0.46 1.91 0.03 1.46
Homog. 0.54 1.38 0.17 2.70 -1.47 1.56
The table reports the DM statistics for a test of the equality of forecast
accuracy for symmetric specifications of the baseline Equation (5) and (10)
with a time-varying relationship. The values were calculated by forecasts for
the months March 1983 to December 1998 for euro area currencies and to
May 2015 for the other ones. The critical values for P → are 1.282, 1.645
and 1.960 for the significance levels 0.10, 0.05 and 0.01, respectively.
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5 - Robustness Analysis
This section details the robustness of the previously presented
results. Some of the robustness issues Rossi (2013) mentioned for exchange
rate studies are already dealt with as 12 different exchange rates and a very
long sample are examined. Nevertheless, issues such as the calibration of the
Kalman filter model or the rolling regressions’ window size remain to be
examined.
5.1 - Calibration of the Kalman Filter Model
In this section an alternative calibration of the Kalman filter model
will be examined. Recall the Kalman filter model of Section 3:
(20)
(21)
where
and with vector including the fundamentals.
Now, the case where is increased to a value of and remains
as is examined. Thus, we use more information of the prediction error
of the model. The results for this modified Kalman filter model are not
presented as it can be said in general that it now performs worse than the
baseline OLS model for all exchange rates. To examine the reasons for this,
Figure 3 shows how the final state estimates for the Pound-US dollar
exchange rate change due to this adjustment of the error term. It can be seen
that the final state coefficient estimates of the adjusted Kalman filter model,
drawn in green, have a larger variance than the baseline Kalman filter or
OLS ones. This is in particular the case for the years between 1995 and
2003. This shows that if the Kalman filter model uses a higher share of the
prediction error as information for the updating of the coefficients, then the
estimates get out of hand. The problem gets worse if is increased further.
Setting smaller than in the baseline Kalman filter model alters the results
only slightly. Changes of have no significant impact.
63
Figure 3: Coefficients Estimated with Different Kalman Filter Models or
OLS
The previous paragraph showed that using too much information of
the prediction error leads to worse results. A reason for this can be that the
Kalman filter cannot distinguish whether the model performs badly because
of a constant relationship between exchange rates and fundamentals, or
missing explanatory information. Compare for example the order flow
model of Evans and Lyons (2002). If the bad performance is mainly due to
the latter, then the Kalman filter should not use the entire information of the
prediction error to estimate the time-varying coefficients.
5.2 - Window Size of the Rolling Regressions
In the exchange rate forecasting literature window sizes are often
chosen arbitrarily. For example, Meese and Rogoff (1983) used a window of
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93 observations for monthly data, Molodtsova and Papell (2009) took 120
observations, and Gourinchas and Rey (2007) used 104 for quarterly data,
while Cheung et al. (2005) took 42 and 59. This can lead to two
shortcomings. Firstly, the ad hoc chosen window may not find a significant
difference between the forecasting ability of the competing models, though it
exists for shorter or longer window sizes. Secondly, the researcher could
compare models for different window sizes until he finds satisfactory results.
Addressing the first point, Inoue and Rossi (2012), using the same data and
methodology as Molodtsova and Papell (2009), found improved evidence for
predictability at smaller windows.
To analyze whether the favorable results of the baseline Kalman
filter model are robust to the chosen window size of the rolling regressions,
the regressions were estimated for windows between 48 and 120. For
comparability, the sample is held constant across window sizes although a
shorter window would allow using a longer sample. The starting and ending
points of the window sizes are chosen such that between 4 and 10 years of
data are incorporated in the regressions.
The results across different window sizes for the Pound-US dollar
exchange rate can be seen in Figure 4. The MSPE of different specifications
of Equations (5) and (10) are grouped in one graph. The Kalman filter
model’s MSPE is drawn in red. For all groups it can be said that smaller
window sizes lead to slightly higher MSPE. To get a clearer view, the DM
statistics for a comparison between Equations (5) and (10) are also shown
for the Pound-US dollar exchange rate in Figure 5. Lines are drawn in the
graphs at -1.282 and 1.282 for the 10% significance levels of the DM
statistic. While for the right hand models the previous observation holds, it is
less clear for the left hand side ones. Therefore, a comparison of Equations
(5) and (10) depends substantially on the chosen window size. The figures,
only for the non-euro countries to save space, can be found in the appendix
beginning with Figure 6. In general, the favorable results for the Kalman
filter are robust to the choice of the window size, and for some currencies
and specifications even strengthened. This result is particularly interesting as
for lower window sizes a time-varying relationship is already more strongly
implied for the baseline model.
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Figure 4: MSPE across Different Window Sizes of Equations (5) and (10)
for the Pound-US Dollar Exchange Rate
Figure 5: DM Statistics across Different Window Sizes of Equations (5) and
(10) for the Pound-US Dollar Exchange Rate
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Figure 6: DM Statistics across Different Window Sizes of Equations (5) and
(10) for the Canadian Dollar-US Dollar Exchange Rate
Figure 7: DM Statistics across Different Window Sizes of Equations (5) and
(10) for the Australian Dollar-US Dollar Exchange Rate
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Figure 8: DM Statistics across Different Window Sizes of Equations (5) and
(10) for the Danish Kroner-US Dollar Exchange Rate
Figure 9: DM Statistics across Different Window Sizes of Equations (5) and
(10) for the Swedish Kronor-US Dollar Exchange Rate
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Figure 10: DM Statistics across Different Window Sizes of Equations (5)
and (10) for the Swiss Franc-US Dollar Exchange Rate
6 - Conclusion
Although exchange rates are very important, it is not yet possible to
perform reliable forecasts. A solution for this puzzle of the economics
profession could be to incorporate a time-varying relationship into exchange
rate models. For example, Sarno and Valente (2009) were able to beat the
random walk in out-of-sample forecasting if they chose the model with the
lowest forecast error out of a selection with all combinations of six different
fundamentals ex post for every forecast. Here it is examined whether
estimating a time-varying relationship with a Kalman filter model using ex
ante information can improve forecasting compared to a model with a
constant relationship. Using a Taylor rule exchange rate model, which is
found to have promising out-of-sample forecasting abilities for the short-
term (Molodtsova and Papell, 2009; Rossi, 2013), it is possible to further
enhance the forecasting performance if the information used to update the
time-varying coefficients is restricted. This is necessary as classic exchange
rate models do not perform badly only due to a constant relationship
between the exchange rate and its fundamentals, but also because of missing
information. See Evans and Lyons (2002) to compare. Thus, without the
restriction of the information, the Kalman filter would adjust the coefficients
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too strongly. We thereby show that the Kalman filter is still a useful
technique for exchange rate forecasting compared to OLS. We further note
for future research that such an adjustment to the calibration could also
improve forecasting with other models which allow for a time-varying
relationship as Markov switching or smooth transition models.
Still, this approach is not able to beat the random walk in forecasting
performance so that the result of Meese and Rogoff (1983) prevails.
However, exchange rate models which incorporate more explanatory
information, such as the order flow model of Evans and Lyons (2002),
would allow an improvement of the estimations of the Kalman filter. Until
now, the Kalman filter cannot distinguish whether a high prediction error is
due to missing information or a constant relationship between the exchange
rate and the fundamentals. Thus, with further explanatory information, the
restriction on the information used by the Kalman filter to estimate the time-
varying coefficients can be eased as not only the prediction error is smaller,
but also because the size of the error is (more) exclusively determined by the
negligence of the time-varying relationship. With this enhanced estimation
of the time-varying coefficients, forecasting accuracy could be improved.
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